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An asymptotic series with respect to the sequence
 
An asymptotic series with respect to the sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a0137501.png" /></td> </tr></table>
+
$$
 +
\{ x  ^ {-n} \} \  ( x \rightarrow \infty )
 +
$$
  
 
or with respect to a sequence
 
or with respect to a sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a0137502.png" /></td> </tr></table>
+
$$
 +
\{ ( x - x _ {0} )  ^ {n} \} \  ( x \rightarrow x _ {0} )
 +
$$
  
 
(cf. [[Asymptotic expansion|Asymptotic expansion]] of a function). Asymptotic power series may be added, multiplied, divided and integrated just like convergent power series.
 
(cf. [[Asymptotic expansion|Asymptotic expansion]] of a function). Asymptotic power series may be added, multiplied, divided and integrated just like convergent power series.
  
Let two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a0137503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a0137504.png" /> have the following asymptotic expansions as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a0137505.png" />:
+
Let two functions $  f(x) $
 +
and $  g(x) $
 +
have the following asymptotic expansions as $  x \rightarrow \infty $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a0137506.png" /></td> </tr></table>
+
$$
 +
f (x)  \sim  \sum _ {n = 0 } ^  \infty 
 +
 
 +
\frac{a _ {n} }{x  ^ {n} }
 +
,\ \
 +
g (x)  \sim  \sum _ {n = 0 } ^  \infty  \frac{b _ {n} }{x^n}.
 +
$$
  
 
Then
 
Then
Line 17: Line 41:
 
1)
 
1)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a0137507.png" /></td> </tr></table>
+
$$
 +
Af (x) + Bg (x)  \sim  \sum _ {n =0} ^  \infty \frac{A a _ {n} + B b _ {n} }{x^n}
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a0137508.png" /> are constants);
+
($A, B $ are constants);
  
 
2)
 
2)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a0137509.png" /></td> </tr></table>
+
$$
 +
f (x) g (x)  \sim  \sum _ {n = 0} ^  \infty 
 +
 
 +
\frac{c _ {n} }{x^n }
 +
;
 +
$$
  
 
3)
 
3)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375010.png" /></td> </tr></table>
+
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375011.png" /> are calculated as for convergent power series);
+
\frac{1}{f(x)}
 +
  \sim 
 +
\frac{1}{a_0} + \sum _ {n = 1 } ^  \infty  \frac{d_n} {x^n} ,\  a _ {0} \neq 0
 +
$$
  
4) if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375012.png" /> is continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375013.png" />, then
+
($c _ {n} , d _ {n} $ are calculated as for convergent power series);
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375014.png" /></td> </tr></table>
+
4) if the function  $  f(x) $
 +
is continuous for  $  x > a > 0 $,
 +
then
  
5) an asymptotic power series cannot always be differentiated, but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375015.png" /> has a continuous derivative which can be expanded into an asymptotic power series, then
+
$$
 +
\int\limits _ { x } ^  \infty  \left ( f (t) - a _ {0} -
 +
\frac{a _ {1} }{t}
 +
\right )  dt
 +
\sim  \sum _ {n = 1 } ^  \infty 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375016.png" /></td> </tr></table>
+
\frac{a _ {n+1} }{n x  ^ {n} }
 +
;
 +
$$
 +
 
 +
5) an asymptotic power series cannot always be differentiated, but if  $  f(x) $
 +
has a continuous derivative which can be expanded into an asymptotic power series, then
 +
 
 +
$$
 +
f ^ { \prime } (x)  \sim  - \sum _ {n = 2 } ^  \infty 
 +
 
 +
\frac{( n - 1 ) a _ {n-1} }{x^n} .
 +
$$
  
 
Examples of asymptotic power series.
 
Examples of asymptotic power series.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375017.png" /></td> </tr></table>
+
$$
 
+
\int\limits _ { x } ^  \infty 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375018.png" /></td> </tr></table>
+
\frac{e  ^ {x-t} }{t}
 +
  dt  \sim \
 +
\sum _ {n = 1 } ^  \infty  \frac{( -1 )  ^ {n-1} ( n - 1 ) ! }{x^n} ;
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375019.png" /> is the Hankel function of order zero (cf. [[Hankel functions|Hankel functions]]) (the above asymptotic power series diverge for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375020.png" />).
+
$$
 +
\sqrt {x } e  ^ {-ix} H _ {0}  ^ {(1)} (x)  \sim  \sum _ {n = 0 } ^  \infty 
 +
\frac{e ^ {-i \pi / 4 } ( - i )  ^ {n} [
 +
\Gamma ( n + 1 / 2 ) ]  ^ {2} }{2  ^ {n-1/2} \pi  ^ {3/2} n ! x  ^ {n} }
 +
,
 +
$$
  
Similar assertions are also valid for functions of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375021.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375022.png" /> in a neighbourhood of the point at infinity or inside an angle. For a complex variable 5) takes the following form: If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375023.png" /> is regular in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375024.png" /> and if
+
where  $  {H _ {0}  ^ {(1)} } (x) $
 +
is the Hankel function of order zero (cf. [[Hankel functions|Hankel functions]]) (the above asymptotic power series diverge for all  $  x $).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375025.png" /></td> </tr></table>
+
Similar assertions are also valid for functions of a complex variable  $  z $
 +
as  $  z \rightarrow \infty $
 +
in a neighbourhood of the point at infinity or inside an angle. For a complex variable 5) takes the following form: If the function  $  f(z) $
 +
is regular in the domain  $  D= \{ | z | > a, \alpha < | { \mathop{\rm arg} }  z | < \beta \} $
 +
and if
  
uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375026.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375027.png" /> inside any closed angle contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375028.png" />, then
+
$$
 +
f (z)  \sim  \sum _ {n = 0 } ^  \infty 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375029.png" /></td> </tr></table>
+
\frac{a _ {n} }{z^n}
  
uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375030.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375031.png" /> in any closed angle contained in D.
+
$$
  
====References====
+
uniformly in  $  { \mathop{\rm arg} }  z $
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.T. Copson,  "Asymptotic expansions" , Cambridge Univ. Press (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Erdélyi,  "Asymptotic expansions" , Dover, reprint (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.T. Whittaker,   G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)</TD></TR></table>
+
as $ | z | \rightarrow \infty $
 +
inside any closed angle contained in $ D $,  
 +
then
  
 +
$$
 +
f ^ { \prime } (z)  \sim  - \sum _ {n = 1 } ^  \infty 
  
 +
\frac{na _ {n} }{z  ^ {n+1} }
  
====Comments====
+
$$
  
 +
uniformly in  $  \mathop{\rm arg}  z $
 +
as  $  | z | \rightarrow \infty $
 +
in any closed angle contained in D.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.G. de Bruijn,  "Asymptotic methods in analysis" , Dover, reprint  (1981)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  E.T. Copson,  "Asymptotic expansions" , Cambridge Univ. Press  (1965)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  A. Erdélyi,  "Asymptotic expansions" , Dover, reprint  (1956)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N.G. de Bruijn,  "Asymptotic methods in analysis" , Dover, reprint  (1981)</TD></TR>
 +
</table>

Latest revision as of 06:35, 14 April 2024


An asymptotic series with respect to the sequence

$$ \{ x ^ {-n} \} \ ( x \rightarrow \infty ) $$

or with respect to a sequence

$$ \{ ( x - x _ {0} ) ^ {n} \} \ ( x \rightarrow x _ {0} ) $$

(cf. Asymptotic expansion of a function). Asymptotic power series may be added, multiplied, divided and integrated just like convergent power series.

Let two functions $ f(x) $ and $ g(x) $ have the following asymptotic expansions as $ x \rightarrow \infty $:

$$ f (x) \sim \sum _ {n = 0 } ^ \infty \frac{a _ {n} }{x ^ {n} } ,\ \ g (x) \sim \sum _ {n = 0 } ^ \infty \frac{b _ {n} }{x^n}. $$

Then

1)

$$ Af (x) + Bg (x) \sim \sum _ {n =0} ^ \infty \frac{A a _ {n} + B b _ {n} }{x^n} $$

($A, B $ are constants);

2)

$$ f (x) g (x) \sim \sum _ {n = 0} ^ \infty \frac{c _ {n} }{x^n } ; $$

3)

$$ \frac{1}{f(x)} \sim \frac{1}{a_0} + \sum _ {n = 1 } ^ \infty \frac{d_n} {x^n} ,\ a _ {0} \neq 0 $$

($c _ {n} , d _ {n} $ are calculated as for convergent power series);

4) if the function $ f(x) $ is continuous for $ x > a > 0 $, then

$$ \int\limits _ { x } ^ \infty \left ( f (t) - a _ {0} - \frac{a _ {1} }{t} \right ) dt \sim \sum _ {n = 1 } ^ \infty \frac{a _ {n+1} }{n x ^ {n} } ; $$

5) an asymptotic power series cannot always be differentiated, but if $ f(x) $ has a continuous derivative which can be expanded into an asymptotic power series, then

$$ f ^ { \prime } (x) \sim - \sum _ {n = 2 } ^ \infty \frac{( n - 1 ) a _ {n-1} }{x^n} . $$

Examples of asymptotic power series.

$$ \int\limits _ { x } ^ \infty \frac{e ^ {x-t} }{t} dt \sim \ \sum _ {n = 1 } ^ \infty \frac{( -1 ) ^ {n-1} ( n - 1 ) ! }{x^n} ; $$

$$ \sqrt {x } e ^ {-ix} H _ {0} ^ {(1)} (x) \sim \sum _ {n = 0 } ^ \infty \frac{e ^ {-i \pi / 4 } ( - i ) ^ {n} [ \Gamma ( n + 1 / 2 ) ] ^ {2} }{2 ^ {n-1/2} \pi ^ {3/2} n ! x ^ {n} } , $$

where $ {H _ {0} ^ {(1)} } (x) $ is the Hankel function of order zero (cf. Hankel functions) (the above asymptotic power series diverge for all $ x $).

Similar assertions are also valid for functions of a complex variable $ z $ as $ z \rightarrow \infty $ in a neighbourhood of the point at infinity or inside an angle. For a complex variable 5) takes the following form: If the function $ f(z) $ is regular in the domain $ D= \{ | z | > a, \alpha < | { \mathop{\rm arg} } z | < \beta \} $ and if

$$ f (z) \sim \sum _ {n = 0 } ^ \infty \frac{a _ {n} }{z^n} $$

uniformly in $ { \mathop{\rm arg} } z $ as $ | z | \rightarrow \infty $ inside any closed angle contained in $ D $, then

$$ f ^ { \prime } (z) \sim - \sum _ {n = 1 } ^ \infty \frac{na _ {n} }{z ^ {n+1} } $$

uniformly in $ \mathop{\rm arg} z $ as $ | z | \rightarrow \infty $ in any closed angle contained in D.

References

[1] E.T. Copson, "Asymptotic expansions" , Cambridge Univ. Press (1965)
[2] A. Erdélyi, "Asymptotic expansions" , Dover, reprint (1956)
[3] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952)
[a1] N.G. de Bruijn, "Asymptotic methods in analysis" , Dover, reprint (1981)
How to Cite This Entry:
Asymptotic power series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotic_power_series&oldid=11585
This article was adapted from an original article by M.I. Shabunin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article